Understanding Conditionalization

Abstract

At the heart of the Bayesianism is a rule, Conditionalization, which tells us how to update our beliefs. Typical formulations of this rule are underspecified. This paper considers how, exactly, this rule should be formulated. It focuses on three issues: when a subject’s evidence is received, whether the rule prescribes sequential or interval updates, and whether the rule is narrow or wide scope. After examining these issues, it argues that there are two distinct and equally viable versions of Conditionalization to choose from. And which version we choose has interesting ramifications, bearing on issues such as whether Conditionalization can handle continuous evidence, and whether Jeffrey Conditionalization is really a generalization of Conditionalization.

Keywords:

• Conditionalization
• Bayesianism
• belief revision
• belief updating
• formal epistemology

Notes

1 Of course, questions of how to understand the concepts the rule employs and questions of how to formulate the rule aren’t entirely distinct. For example, different choices regarding how to think of credences can have a bearing on the logical form of the predicate one uses to represent (say) having a certain credence in a proposition. Likewise, different choices regarding how to formulate the rule can make certain views regarding how to think about credences more or less attractive. But, the discussion that follows will be focused primarily on questions of logical form.

2 The ‘virtually’ caveat is required because of the potential gap between having a credence of 1 in something and being certain of it. (E.g. one should have a credence of 1 that a countably infinite number of fair coin tosses will land tails at least once, but one shouldn’t be certain of this.)

3 That is, a subject’s credences should assign values to propositions that form a Boolean algebra – a set of propositions closed under conjunction and negation – and the values it assigns should satisfy the (finite) probability axioms: (i) , (ii) (where is the trivially true proposition), (iii) if A and B are mutually exclusive, then .

4 Of course, there are a number of different formulations of Conditionalization in the literature to choose from. I’ve chosen this one because it leaves all three of the questions I’ll be exploring open. In order to figure out how to best formulate Conditionalization, we need to assess the different possible answers to these questions. And, a formulation of Conditionalization which leaves all three of these questions open, like the one given above, provides us with an ideal place to start.

5 In the next section, we’ll be considering whether we should be thinking of E as something you get at a particular time, or as the cumulative evidence one receives over some interval of time. In the latter case, I’ll undertstand ‘t(E)’ as the final time over which one gets E; i.e. the future endpoint of this interval.

6 I’m assuming in the text that there’s a time t(E) at which one gets E as evidence and a time at which one should first adopt . But, this needn’t be the case. It could be that the temporal interval during which one has E as evidence is an open interval, so that there is no initial time at which E is received. Likewise, it could be that the temporal interval during which one should have credences in an open interval, so that there is no first time at which should be adopted. Although these possibilities complicate things, they don’t end up changing the dialectic in any interesting way (see footnote ). So, I’ll put these possibilities aside.

7 The formulations of Conditionalization given in the literature vary widely with respect to this question: some assume the posterior answer (e.g. Earman 1992, Howson and Urbach 2006), some assume the concurrent answer (e.g. (e.g. Lewis 2010 and Strevens 2015), and some are silent on the matter (e.g. Easwaran 2011 and Weisberg 2011).

8 I’m assuming here that the time at which a subject ‘gets’ evidence is the time at which the evidence becomes relevant to our epistemic evaluation of the subject. But I do not take this to be controversial, since this assumption is shared by virtually everyone. (Consider: if this were not the case, then the oft repeated truism that a subject’s beliefs should take all of her evidence into account (the so-called ‘Principle of Total Evidence’) would be untenable. If, for example, one characterized ‘receiving evidence’ such that the time at which a subject received visual evidence was one minute (or one year, or one century) before the light struck her eyes, it would be implausible to say that she should take all of her evidence into account.)

9 See Lewis (1980).

10 Other proponents of this kind of stance regarding epistemic norms include Feldman (2001) and Wolterstorff (2010).

11 One might reasonably want to hear more about what, exactly, it means to say that Conditionalization is ‘an ideal at which to aim’ or an ideal performance norm’, and about how this understanding of Conditionalization interacts with things like a subject’s cognitive capabilities and ought-implies-can. I describe one natural way of spelling out these notions and their interaction, using the framework of Kratzer (1991), in Appendix B.

12 Of course, this kind of worry won’t arise given certain natural pictures of what receiving and updating on evidence is like. For example, suppose one adopts an account of evidence (such as Howson and Urbach’s 1993 or Williamson’s 2000) according to which receiving evidence E is, at least in part, a matter of coming to believe E. And, suppose one takes such belief changes to be global and concurrent. That is, just as stepping on to a trampoline changes the elevation of both the place one’s standing and the surrounding area, getting E as evidence changes both one’s credence in E and one’s credence in the ‘surrounding’ propositions. And just as the full change in elevation of the place one steps doesn’t happen before any of the other changes in elevation take place – the changes in elevation of different parts of the trampoline are (roughly) concurrent – getting E as evidence doesn’t take place before these other belief changes take place, these belief changes are concurrent. On this picture, Conditionalization is naturally thought of as describing what the shape of these global concurrent belief changes should be like. And, the worries regarding potential conflicts between instantaneous updating and procedural norms described in the text won’t arise on such a picture.

13 In making this reply, I assume that in order for a set of norms to provide a coherent standard of ideal performance, or a coherent ideal for us to aim at, it only has to be logically possible to satisfy them. But one might wonder why these ideals shouldn’t also have to be metaphysically possible to live up to. And if they do, and if instantaneous causation is metaphysically impossible, then it seems that these natural procedural norms and the concurrent understanding of Conditionalization are in conflict after all. (Having the right beliefs (those prescribed by Conditionalization) in the right way (via some causal process initiated by the receipt of evidence) seems to require instantaneous causation, since one’s beliefs must instantly change in light of one’s evidence. Thus, if instantaneous causation is metaphysically impossible, then so is jointly satisfying Conditionalization and these procedural norms. And if jointly satisfying Conditionalization and these procedural norms has to be metaphysically possible in order for these norms to be jointly true, then they can’t be jointly true.)6 So why do I require these ideals to be logically possible, but not metaphysically possible? Here is why. For something to usefully serve as an ideal at which to aim, it needs to be something which we can approach by degrees; something for which we can discern paths of states that lead to it such that each state along the path gets closer to satisfying the ideal. Thus, metaphysically impossible norms can serve as useful ideals at which to aim: one can make sense of moving toward or away from the ideal of updating instantly by updating more or less quickly. But it’s hard to see how logically impossible ideals could serve as useful ideals at which to aim: it’s not clear how one could move closer or father away from the ideal of being a round square, or a married bachelor.

14 As noted in footnote 6, I’ve simplified this discussion by assuming that there are particular times t(E) and at which a subject receives her evidence and should adopt her new credences. But this needn’t be the case (e.g. if the period during which a subject should have is an open interval). Introducing these possibilities complicates the dialectic in two ways, but these complications end up effectively canceling each other out. The first complication is that these possibilities leave us with three natural ways to group answers to the question of how the receipt of E and the adoption of are related:

(1)	E is received before is adopted, and there is a gap between the receipt of E and .
(2)	E is received at the same time as is adopted, and thus there’s no gap between them.
(3)	E is received before is adopted, but there is no gap between the receipt of E and (e.g. E’s received at some time t, and is adopted at the open endpoint of an interval starting at t).

(1) and (2) correspond to the posterior and concurrent answers considered in the text, while (3) is a possibility which only comes into view once we drop the simplifying assumption. The second complication is that these possibilities allow us to see that the worries raised for the different answers track slightly different issues. The worries regarding total evidence and fit with Howson-and Urbach-like pictures of evidence arise for any view on which E is received before is adopted (thus applying to (1) and (3)). The worries regarding ought implies can and substantive and procedural requirements arise for any view on which there’s no gap between the receipt of E and the adoption of (thus applying to (2) and (3)). But together these complications allow us to see that no interesting positions are left out by simplifying and ignoring (3). For (3) is strictly less appealing than (1) and (2), as it is subject to the worries raised for both.

15 Of course, it’s a well known feature of Conditionalization that conditionalizing on E and then on F yields the same result as conditionalizing on the conjunct . So one could obtain the result either way. But, on this understanding of the rule, what’s really going on ‘under the hood’ is a sequence of updates; updating on the conjunction is merely a convenient calculational short cut.

16 The notion of ‘cumulative evidence’ is best thought of as one of the basic notions that this interval updating understanding of conditionalization employs. (I.e. whereas the ‘sequential updating’ picture takes as basic a notion of getting a piece of evidence at a particular time, the ‘interval updating’ pictures takes as basic a notion of getting cumulative evidence over an interval.) Of course, when discussing Conditionalization, we have the option of understanding the cumulative evidence a subject receives during an interval as the logically weakest proposition that entails all of the evidence she receives during that interval (or, equivalently, the conjunction of all of the evidence she receives during that interval). This would allow both sequential and interval updating understandings of Conditionalization to take as basic the notion of getting a piece of evidence at a particular time. But when we turn discuss Jeffrey Conditionalization, in Section 4.1, we lose the option of providing a reductive understanding of the notion of ‘cumulative evidence’, since Jeffrey Conditionalization deals with weighted evidence partitions to which the notions of entailment and conjunction don’t apply.

17 The formulations of Conditionalization given in the literature vary widely with respect to this question: some assume the sequential answer (e.g. Earman 1992; Howson and Urbach 2006), others assume the interval answer (e.g. Lewis 2010), and yet others are silent on the matter (e.g. Earman 1992; Weisberg 2011; Strevens 2015).

18 Though see Christensen (1992) for a notable dissent: ‘[Jeffrey Conditionalization] is thus not simply an elegant generalization of [Conditionalization], a pure improvement which merely removes some gratuitous idealization. It removes idealization, but at a price. The additional cases covered by the liberal model are not covered in the same way; and consequently, the account as a whole must be given a different philosophical interpretation’ (547).

19 A number of people, including Levi (1967), Carnap (in Jeffrey 1975), Field (1978), Christensen (1992) and Lange (2000), have maintained that what evidence partition a subject receives should depend on what the subject’s credences are. But they’ve argued for this under the assumption that it’s a substantive question left open by the formalism. And thus they’ve appealed to various intuitive and epistemic considerations to make their case. If the formalism of Jeffrey Conditionalization itself required evidence to be credence-dependent, then the kinds of considerations that Christensen and others have offered in support of this claim would be superfluous. No interesting discussion needs to take place to establish that evidence is credence-dependent if the formalism itself entails it.

20 For discussions of standard deontic logic, see Aqvist (2002) and McNamara (2010).

21 See Feldman (1986) for some reasons for wanting to have this kind of subject and time sensitivity in the context of ethics.

22 A different (and more powerful) way to allow for such variations is to take deontic operators to range over centered worlds instead of worlds. But since this approach raises complications orthogonal to the issues at hand, I employ the more traditional approach of subject and time indexing the deontic operators in the text.

23 Recognizing the time-indexed nature of the obligation operator could be seen as raising a fourth question, in addition to the three questions discussed in the text: a ‘Time of Obligation’ question, regarding what times the obligation operators should be indexed to. One answer is that they’re indexed to the initial time we’re considering (), another is that they’re indexed to the final time we’re considering (), and a third is that they’re indexed to all times – the norm requires a subject to satisfy this constraint at every time, regardless of the times the prescription involves. I tentatively favor the second and third answers over the first, since according to the first answer, it will never be the case that a subject’s obligated to adopt at the time at which she’s supposed to adopt it – by the time rolls around, she’ll only have obligations to adopt some further credences in the future. And I adopt the second answer (instead of the third) in the text because the most plausible version of the narrow scope answer, which I’ll be arguing against in section 6, requires this time to be . (Looking ahead: we need the time to be in order to ensure that the antecedent is true at all C-accessible worlds and yet the consequent is not.) So adopting the second answer allows me to stack the deck in favor of my opponents.

24 In linguistics, the standard framework for formalizing claims involving modals is the framework developed by Kratzer (1991). Although the discussion in the text assumes we’re using standard deontic logic, everything I say can be translated into Kratzer’s more sophisticated framework. Indeed, moving to Kratzer’s framework offers some benefits, for it provides us with the tools to spell out several things that it’s difficult to flesh out using standard deontic logic. Since working out these details takes a bit of time, I’ve relegated my discussion of how to set things up using Kratzer’s framework to Appendix B.

25 See Broome (1999).

26 In the literature, Conditionalization is generally presented in a way that is neutral with respect to this question (such as in Earman 1992; Howson and Urbach 2006; Lewis 2010; Easwaran 2011; Weisberg 2011; Strevens 2015).

27 Broome (2007) shows that changing a norm from narrow to wide scope or vice versa won’t change whether the actual world is one of the best worlds. Given this, one might worry about whether there’s anything substantive at stake here. But these two answers are logically distinct. (E.g. if A is false and at all of the best worlds A is true and C is false, then the narrow scope conditional will be true but the wide scope one false; while if A is false and at all of the best worlds A and C are false, then the narrow scope conditional will be false but the wide scope conditional true.) And, as we’ll see, these two answers will have importantly different implications.

28 For some of the recent literature bearing on this issue, see Broome (1999), Schroeder (2004), Broome (2007), Kolodny (2007), Bedke (2009), Brunero (2010), Way (2011), Brunero (2012), Lord (2013), Shpall (2013), and Titelbaum (forthcoming). (The reason it’s fundamental rational requirements that are of interest is because one might be able to derive some wide scope norms from narrow scope norms or vice versa. So the interesting claims aren’t whether there are any wide/narrow scope rational requirements, but rather whether the are any fundamental (i.e. non-derivative) wide/narrow scope rational requirements.)

29 For example, in the next section, certain questions regarding how we understand Conditionalization (such as how we answer the Time of Evidence question) will bear on the plausibility of wide vs. narrow scope understandings of the rule. We wouldn’t expect the same dialectic to play out for other norms – the shape of this dialectic is particular to Conditionalization.

30 For examples of arguments in favor of narrow scope understandings of fundamental rational requirements, see Schroeder (2004), Kolodny , (2009). It’s worth noting that the terms ‘narrow scope’ and ‘wide scope’ are used somewhat equivocally in this literature; see Titelbaum (forthcoming). But everyone calls norms of the form narrow scope, and norms of the form wide scope, so we can skirt these complications here.

31 For discussions of these kinds of symmetry arguments, see Way (2011), Brunero (2012), Lord (2013) and Shpall (2013).

32 In the text, I focus on worries regarding whether the narrow answer has the right normative profile. For further kinds of arguments against narrow scope understandings of rational requirements, see Brunero (2010) and Shpall (2013).

33 In particular, this proposal follows many of the ideas regarding how to set up deontic structure laid out in Feldman (1986), chapter 2.

34 The notion of ‘C-accessibility’ is the same as the notion of accessibility described in Feldman (1986), chapter 2.

35 Though see Feldman (1986), Section 2.1.1 for some reasons against adopting such a constraint.

36 The reason we need to assume the posterior answer is that it ensures there’s a temporal gap between the events described in the antecedent A (having credences cr and receiving evidence E) and the events described in the consequent C (adopting credences ). And this temporal gap is crucial to getting the maneuver described in the text to work. (Briefly: we need A to be in the past to ensure that A is true at all best worlds but only trivially obligatory, and we need C to not be in the past to ensure that C isn’t trivially obligatory.)

37 It’s important to not confuse this result with the result shown by Broome (2007) and discussed in footnote . Broome’s result is both broader and weaker. Broome shows that, in general, switching narrow and wide scope formulations of a rule won’t change whether the actual world is one of the best worlds. This result yields the stronger conclusion that the narrow and wide scope formulations will logically entail each other, but only in a narrower class of cases (those in which the norm is Conditionalization, and the assumptions we’ve been making hold).

38 In a similar fashion, this equivalence undercuts Kolodny’s (2007) ‘Problem of Conflict’ for wide scope understandings of Conditionalization, the worry that ‘[s]ome requirements of formal coherence not only are not explained by a concern for the true and the good, but moreover would forbid what that concern requires’. (Kolodny 2007, 231) This kind of argument cannot support the narrow over wide scope understandings of Conditionalization, since the two understandings make the same prescriptions, and thus ‘forbid’ the same things. Likewise, it undercuts Kolodny’s (2007) ‘Problem of Normativity’ for wide scope understandings of Conditionalization, the worry that there’s no plausible explanation for how wide scope rational requirements could have normative force. Since the narrow and wide scope understandings of Conditionalization are equivalent, any considerations that one could use to justify one of these sets of prescriptions could also be used to justify the other.

39 See Brunero (2010) for a more general version of this worry for narrow scope norms.

40 This assumes that we don’t adopt the additional assumptions sketched above. If we do adopt those assumptions, then the wide answer will yield the same undesirable results as the narrow answer (as one would expect, since given these assumptions the narrow and wide answers are equivalent).

41 I’d like thank the May 2014 UMass Brown Bag Presentation group and the audience of the 2015 Belief, Rationality and Action over Time conference for helpful comments and discussion. In addition, I’d like to thank Jennifer Carr for flagging the third worry for the narrow answer discussed in Section 6.1, Brian Hedden for discussion about the ways of thinking about evidence mentioned in footnote , Sarah Moss and Ralph Wedgewood for discussion about Kratzer semantics and the issues discussed in Appendix B, Miriam Schoenfield for suggesting the names for the answers to the Time of Evidence Question, and for pushing me to address the issues discussed in footnote , and Michael Titelbaum for pushing me to get clearer on what one might mean by things like ‘an ideal at which to aim’ (which I now try to do in Appendix B). Finally, I owe special thanks to Lisa Cassell, Maya Eddon, and Alejandro Perez-Carballo, for detailed comments on the entire paper, which led to more substantive improvements than I could reasonably list.

42 I think these are plausible assumptions for the ordering base corresponding to the notions of epistemic obligation and permission. But not everyone would agree. For example, Christensen (2007) argues that there are inconsistent epistemic ideals.

43 For simplicity, I talk here as if we were taking the modal base and ordering source to be functions from worlds to set of propositions, and to not be indexed to anything. But I think our final account will want these functions to be indexed to subjects and times (or to be functions from centered worlds); see Section 5.

44 To see that these two thoughts are equivalent: given the assumption about the modal base, the formulations of Conditionalization given in the text will entail that at every ideal world subjects conditionalize (in the manner specified by that formulation), and thus that the ordering source must entail that subjects conditionalize. Going the other way, if the ordering source entails that subjects conditionalize (in the manner corresponding to some formulation), then it follows that subjects conditionalize at every ideal world, and thus (given the assumption about the modal base) that subjects should conditionalize, which is just what the corresponding formulation of Conditionalization asserts.

45 Of course, none of the formulations of Conditionalization we’ll consider specify what this range of weaker propositions is. But this isn’t something we should expect from Conditionalization – spelling out how to best decompose the proposition that subjects conditionalize into these weaker claims is a task which requires a lot more information than a simple rule like Conditionalization could provide.

46 The wide scope rule requires that be true at all O-accessible worlds. If O(A) is true, then the first part of that disjunct () will be false at all O-accessible worlds, and thus the second part of the disjunct (C) must be true at all O-accessible worlds. Thus O(C) is true.